The cross spectral density (CSD) can be decomposed into orthogonal components via the  Karhunen-Loeve expansion, also known as the Proper Orthogonal Decomposition Theorem (POD). This allows different features of the field to be separated. It is more general than the wavenumber-frequencyspectrum and is used when the field is not homogeneous. The eigenvalue, l, are the amplitude or energy content; the eigenvectors, y, are the signal form. This decomposition reduces to the standard wavenumbers when the field is homogeneous and the matrix is Toeplitz. The example to the right is clearly inhomogeneous and shows a propagating wave because the real and imaginary components at 90 degrees out of phase.

The decomposition is called a subspace filter when the user chooses how  to truncate the series to reject noise and retain coherence.  A reduced matrix is reconstructed including only the components deemed important.

Examples where the use of the Karhunen-Loeve expansion is valuable include:

• jet noise source location
• filtering transonic wind tunnel noise
• turbulence research into coherent structures